Robust And Sparse Portfolio Model For Index Tracking

In the context of index tracking, the tracking error measures the difference between the performance of the stock market index and the portfolio weights. Several optimization approaches for portfolio selection have been pro-posed to find optimal portfolio weights to reduce the tracking error. In this paper, we use the l2and lp (0<p<1) norm penalties to the tracking portfolio model in order to get a robust and sparse portfolio for index tracking. The l2norm reg-ularizes the over-determined system to impose smoothness to alleviate the effect of the existence of highly correlated variables and hence has better out-of-sample performance and the lp (0<p<1) norm penalty achieves sparsity to account for transaction costs. We enroll in the model explicitly the non-negative constraints, that is, the short-sale constraints appear in practice. The lp (0<p<1) norm penalty is non-Lipschitzian, non-convex which leads to computational difficulty. We adopt the smoothing projected gradient (SPG) method to solve the robust and sparse portfolio model. We show that there exists at least one accumulation point generated by the SPG algorithm for solving this model and any accumula-tion point of the SPG method is a Clarke stationary point. In this paper, this methodology is implemented on two real stock data sets constructed by S&P500and Russell3000, and we find our proposed regularized l2and lp norm penalties portfolio model outperform the other methods.